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Secant varieties of Grassmann Varieties

M.V.Catalisano, A.V.Geramita, A.Gimigliano

Introduction:

The problem of determining the dimensions of the higher secant varieties of the clas-

sically studied projective varieties is a problem with a long and interesting history. The

recent work by J. Alexander and A. Hirschowitz (see [AH], for example) completed a

project that was underway for over 100 years (see [Pa], [Te], and [W]) and confirmed the

conjecture that, apart from the quadratic Veronese varieties and a (few) well known ex-

ceptions, all the Veronese varieties have higher secant varieties of the expected dimension.

There has been no comparable success with the case of the Segre varieties (nor is

there even a compelling conjecture) although there is much interest in this question -

and not only among geometers. In fact, this particular problem is strongly connected

to questions in representation theory, coding theory and algebraic complexity theory (see

our paper [CCG] for some recent results as well as a summary of known results, and

[BCS] for the connections with complexity theory). In this paper we investigate this same

problem for another family of classically studied varieties, namely the Grassmann varieties

in their Plucker embeddings. The dimensions of all the higher secant varieties to the

Grassmannians of lines in projective space are well known (and we give a simple proof of

this known result in Section 2) but, to the best of our knowledge, little more can be found

in the classical or modern literature about this problem (although, see the comments of

Ehrenborg in [E], and [No] for connections with coding theory).

In this paper we give some general results about the dimensions of the higher secant

varieties to the Grassmann varieties. As a corollary we can calculate the dimensions of

the chordal (or secant line) varieties to all the Grassmann varieties. Our result shows that

(apart from the known cases) all these secant varieties have the expected dimension.

In Section 3 we discuss our search for deficient secant varieties to Grassmannians. We

give new proofs that G(3, 7) and G(3, 9) have deficient secant varieties and also find a new

Grassmannian with deficient secant varieties, namely G(4, 8). Finally some words con-

cerning our methods. Our approach uses, as its first step, the fundamental observation of

Terracini about secant varieties (the so-called Lemma of Terracini). This Lemma converts

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the problem of determining the dimension of a (higher) secant variety into one of finding

the dimension of a certain linear space. The second step in our approach is to identify

this linear space (via an exterior algebra version of apolarity) with a graded piece of an

intersection of homogenous “fat ideals” in an exterior algebra. The final step consists of

calculating the dimension of the appropriate piece of this ideal in the exterior algebra.

1. Preliminaries, Grassmannians and Exterior Algebras.

We recall a few elementary facts about exterior algebras and Grassmannians. Let V be

a finite dimensional vector space over the field K (we will always assume that charK = 0

and that K is algebraically closed) and let dim V = n+1. Then∧

(V ) denotes the exterior

algebra of V .∧

(V ) is a graded (non-commutative) ring such that (∧

(V ))k = ∧kV .

It is well-known that ∧k(V ) is a finite dimensional vector space and that dim∧k(V ) =(n+1

k

). The elements of ∧k(V ) are called exterior k-vectors (and sometimes skew-symmetric

tensors). An exterior k-vector T is said to be decomposable if we can find vectors v1, ..., vk ∈V such that T = v1 ∧ ...∧ vk. We say that the exterior k-vector T has ∧-rank = r if it can

be written as a sum of r (but no fewer) decomposable k-exterior vectors.

It is natural to ask the following two questions:

1) What is the least integer D(k, n + 1) so that every exterior vector

in ∧k(V ) has ∧-rank ≤ D(k, n + 1)?

2) What is the least integer E(k, n + 1) for which there is a dense

subset U ⊂ ∧k(V ) (dense in the Zariski topology) such that every

exterior vector in U has ∧-rank ≤ E(k, n + 1)? We will call this

number the typical rank (called the essential rank in [E]) of ∧k(V )?

Our main focus in this paper will be on Question 2).

We first recall the coordinate description of the canonical multilinear, alternating map

νk : V × · · · × V︸ ︷︷ ︸

k−times

: → ∧k(V )

(v1, ..., vk) → v1 ∧ ... ∧ vk

.

Choose an ordered basis {e0, ..., en} for V . We identify the elements of V × · · · × V︸ ︷︷ ︸

k−times

with k × (n + 1) matrices having entries in K by writing vj =∑n

i=0 ai,jei and letting

-2- 2/1/08

M = (ai,j) be the k × (n + 1) matrix which has the coordinates of the vector vj as its jth

row. We will assume, from now on, that we have chosen such an ordered basis for V and

that we are making the identification above.

With the basis for V as above, it is standard to choose, as an ordered basis for ∧kV ,

the set

{ei1 ∧ · · · ∧ eik| 0 ≤ i1 < · · · < ik ≤ n}

where the subsets {i1, . . . , ik} are ordered lexicographically. With this choice the map νk

can now be written

νk(v1, . . . , vk) =∑

{i1,...,ik}mi1,...,ik

ei1 ∧ ... ∧ eik

where mi1,...,ikis the k × k-minor of M formed by the columns i1, i2, . . . , ik.

It is clear from this description that v1 ∧ ... ∧ vk 6= 0 if and only if the matrix M has

rank k i.e. if and only if {v1, . . . , vk} are a linearly independent set of vectors if and only

if 〈v1, . . . , vk〉 (the linear space spanned by the vectors v1, . . . , vk ) is k-dimensional.

If {v′1, . . . , v

′k} is another set of k vectors from V , and {v′

1, · · · , v′k} corresponds to the

k × (n + 1) matrix M ′, then it is a standard fact of linear algebra that

〈v1, . . . , vk〉 = 〈v′1, . . . , v

′k〉 ⇔ ∃ an invertible k × k matrix A such that M = AM ′.

It follows that, in the notation above,

mi1,...,ik= (det A)m′

i1,...,ik. (†)

These last considerations suggest that:

1) we consider the map νk as being defined on the quotient space

of V × · · · × V︸ ︷︷ ︸

k−times

given by the action of the group GLk and, in par-

ticular, only on those elements of the quotient space which have, as

representative, a set of k linearly independent vectors (equivalently, is

represented by a k× (n+1) matrix of rank k), i.e. we should consider

the map νk as being defined on the Grassmannian of k-dimensional

subspaces of Kn+1 ≃ V (denoted Gk,n+1);

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2) we consider the target space of νk as P(∧k(V )) ≃ PNk (where

Nk =(n+1

k

)− 1) and we consider the coordinate functions on P

Nk to

be zi1,...,ikwhere the sets {i1, . . . , ik} are chosen as above.

Remarks: 1) None of what we have described above is new, i.e. this is the usual description

of the Plucker embedding of the Grassmannian of k-dimensional subspaces of an n + 1-

dimensional vector space into a projective space. We have insisted on recalling all the

standard ideas of this embedding because we will be using the explicit description of this

embedding later and we want to have all the notation present for the reader.

2) Once we choose a basis for V there is a natural 1-1 correspondence between the asso-

ciated bases of ∧kV and ∧n+1−kV . It is well known that, if we use this corredspondence

as an identification, the equations defining the Plucker embeddings of ∧kV and ∧n+1−kV

into PNk = P

Nn+1−k are the same. Thus, for the remainder of this paper we will only

consider the Grassmanians Gk,n+1 where k ≤ [(n + 1)/2].

We will have occasion to use another standard fact from linear algebra, namely: two

matrices of size r × s have the same row space if and only if their row reduced echelon

forms are equal.

The way we will use this fact is that we will consider, as representatives of the Grass-

mannian, Gk,n+1, those matrices of size k × (n + 1) which have rank k and which are in

row-reduced echelon form. By the comment above, these are precisely the matrices which

correspond to the different k-dimensional subspaces of V and on which the map νk is now

being defined.

Definition 1.1: Let X ⊆ PN be a closed irreducible projective variety; the sth (higher)

secant variety of X is the closure of the union of all linear spaces spanned by s points of

X . We will use Xs to denote the sth secant variety of X .

There is an “expected dimension” for Xs: i.e. if dim X = n, one “expects” that

dim Xs = min{N, sn+ s− 1}. This estimate comes from observing that choosing s points

from X gives sn parameters and, having made that choice, we are looking at points on a

projective space of dimension s − 1. Thus there are sn + (s − 1) parameters, assuming all

choices were independent. Of course, the secant varieties to a variety already embedded in

PN cannot have a dimension which exceeds N . Putting these two things together gives the

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“expected” dimension for the variety Xs that we wrote above. Since it need not always be

the case that the parameters mentioned above are independent, it is not always the case

that Xs has the “expected dimension”. In such a case we have dim Xs < min{N, sn+s−1}and we say that Xs is defective. A measure of this “defectiveness” is given by the quantity

min{N, sn + s − 1} − dimXs.

Let us go back to the embedded Grassmannian, νk(Gk,n+1) ⊂ PNk . Since this variety

parameterizes the decomposable exterior vectors in ∧k(V ), its secant varieties [νk(Gk,n+1)]s

can be viewed as the closure of the locus of exterior vectors having ∧-rank = s. Hence

we have another interpretation of the numbers E(k, n + 1) we introduced above. More

precisely:

Fact: Let V be a K-vector space with dimK V = n + 1, then:

E(k, n + 1) = min{s | [νk(Gk,n+1)]s = P

Nk}.

Moreover, information about the dimension of [νk(Gk,n+1)]s, for values of s different from

E(k, n + 1), will tell us about the stratification of P(∧kV ) with respect to ∧-rank.

Before proceeding it is worthwhile to recall that the dimensions of all the secant

varieties to the Grassmannian of 2-dimensional subspaces of an n + 1-dimensional space,

i.e. the dimensions of [ν2(G2,n+1)]s, are known (e.g. see [Z] ). Also, some results on

E(k, n + 1) (from a more algebraic and combinatorial point of view) can be found in [E].

One of the main tools we will use to find the dimensions of secant varieties is the

famous Lemma di Terracini. We recall that lemma now.

Lemma: (Terracini’s Lemma) Let X be an irreducible variety in Pn and let P1, . . . , Ps be

generic points of X . Let TPibe the projectivized tangent space to X at Pi and denote by

〈TP1, . . . , TPs

〉 the (projective) linear subspace of Pn spanned by the TPi

. Then,

dim Xs = dim〈TP1, . . . , TPs

〉 .

In view of Terracini’s Lemma, we need a good description of the tangent space to a

point of νk(Gk,n+1) ⊂ PNk . We do this by studying the differential of the map νk above.

Since it is immaterial which point of νk(Gk,n+1) we consider, we will consider the point

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νk(M), for M = ( Ik 0 ), where Ik is the k×k identity matrix and 0 is the k× (n+1−k)

matrix of zeroes.

The image of this point of the Grassmannian is the point [1 : 0 : . . . : 0] of the Plucker

embedding in PNk . This is a point in the affine piece of P

Nk which is the complement

of the closed set V (z0,1,2,...,k−1), i.e. it is the point in ANk whose affine coordinates are

(0, 0, . . . , 0).

The points of the Grassmannian, Gk,n+1, with image in this affine piece are all rep-

resented by matrices in row reduced echelon form of the type ( Ik A ), where A is any

matrix of size k × (n + 1 − k). Thus, an affine version of the map νk can be described by:

νk : Ak(n+1−k) −→ P

Nk\V (z0,1,2,...,k−1) ≃ ANk = A

(n+1

k )−1

where, if A is the k × (n + 1 − k) matrix which represents a point of Ak(n+1−k), then the

coordinates of the image of A are given by considering all the maximal minors of ( Ik A )

except the first minor, i.e the minor corresponding to the first k columns.

We now want to compute the linear transformation dνk(M) explicitly. Since we know

that νk(Gk,n+1) is a smooth variety of dimension k(n + 1− k), we are not so interested in

the rank of this linear transformation (it has to be k(n + 1 − k)). Rather, what we need

is an explicit description of the vectors in the image of this linear transformation.

Since M = ( Ik 0 ) corresponds to the origin of Ak(n+1−k), the tangent space to M

can be thought of as all the matrices B of size k × (n + 1 − k). A curve in Ak(n+1−k)

through M with tangent vector B at M is given by the line λB, where λ ∈ K. The image

(under νk) of a particular point on this line (call it λ0B) is given by the k × k minors of

the matrix ( Ik λ0B ) (not the first minor). The minors which involve all but 1 column of

Ik give us all the possible λ0bi,j, where bi,j runs through all the entries of B, while those

minors which involve all but r columns of Ik give us λr0(∗) where ∗ is some r × r minor of

B. Thus the image of B under the differential dνk(M) is:

dνk(M)(B) = limλ→0(1/λ)(νk(λB) − νk(0))

It is easy to see that, in the limit, we get the entires bi,j , wherever there was a λbi,j,

and a 0 wherever there was a λr(∗) for r > 1. Put another way, the affine cone over

Tνk(M)(νk(Gk,n+1)) is:

{(. . . , ci1,...,ik, . . .) ∈ A(n+1

k ) | where ci1,...,ik= 0 if more than one of

i1, . . . , ik is different from 0, 1, . . . , k − 1

}

.

-6- 2/1/08

I.e. it is the vector subspace of ∧k(V ) generated by all the ei1 ∧ . . . ∧ eikwhere at least

(k − 1) of the ij ’s are in {0, 1, . . . , k − 1}.

Recall that there is a natural “apolarity” in the exterior algebra∧

(V ), i.e. there is a

perfect pairing

∧k(V ) × ∧n+1−k(V ) −→ ∧n+1V ≃ K

induced by the multiplication in∧

(V ).

Thus, if Y is any subspace of ∧k(V ), we can associate to Y , its perpendicular space

Y ⊥ ⊆ ∧n+1−k(V ) where

Y ⊥ := {w ∈ ∧n+1−k(V ) | v ∧ w = 0 for all v ∈ Y }.

Of course, (Y ⊥)⊥ = Y and from standard facts of linear algebra we have

dimK Y ⊥ + dimK Y = dimK ∧k(V ) = dimK ∧n+1−k(V ) .

Now, if we let Y = Tνk(M)(νk(Gk,n+1)) (with M as above) then

Y ⊥ = 〈ej1 ∧ . . . ∧ ejn+1−k| at least two of {j1, . . . , jn+1−k} are in {0, 1, . . . , k − 1}〉.

Put another way,

Y ⊥ = [(e0, . . . , ek−1)2]n+1−k (∗)

i.e. Y ⊥ is the degree n + 1 − k part of the square of the ideal of∧

(V ) generated by

e0, . . . , ek−1.

Since, quite generally, whenever W1, . . . , Ws are subspaces of ∧k(V ) we have that

(W1 + · · ·+ Ws)⊥ = W⊥

1 ∩ · · · ∩ W⊥s

we obtain, by applying Terracini’s Lemma and the results just obtained above, that:

Proposition 1.2: Let V be a vector space of dimension n + 1 and let

B1 = {v1,1, . . . , vk,1}, · · · ,Bs = {v1,s, . . . , vk,s}

-7- 2/1/08

be a collection of s sets of k generic vectors in V . Let Ij = (v1,j, . . . , vk,j) ⊂∧

(V ),

j = 1, . . . , s and let

W = (I12 ∩ . . . ∩ Is2)n+1−k

Then the dimension of [νk(Gk,n+1)]s is

dimKW⊥ − 1 =

[(n + 1

k

)

− dimKW

]

− 1 =

[(n + 1

k

)

− 1

]

− dimKW

. ⊓⊔

If the subspaces Vi, spanned by the Bi above, are as “disjoint as possible”, we expect

that

dimK W = max{(

n + 1

k

)

− s[k(n + 1 − k) + 1], 0}

i.e. that

dim[νk(Gk,n+1)]s = min{

(n + 1

k

)

− 1, s[k(n + 1 − k)] + (s − 1)}

which is what we called the expected dimension of [νk(Gk,n+1)]s. I.e. if W has the

expected dimension then so does [νk(Gk,n+1)]s. Moreover, if W has more than the expected

dimension then the difference is precisely what we called the defectiveness of [νk(Gk,n+1)]s.

Remark: Notice that as long as ks ≤ n + 1 we can choose the vectors vi,j to be part of

the basis {e0, . . . , en} of V . It is precisely this case we will consider in the next section.

2. The “monomial” case.

In this section we will suppose that ks ≤ n + 1 and let

W = [(e0, ..., ek−1)2 ∩ ... ∩ (eks−k, ..., eks−1)2]n+1−k.

By analogy with the case of ideals in the symmetric algebra of a free module, we call call

such ideals ”monomial ideals” of the exterior algebra. Thus, we can view W as the degree

n + 1 − k part of a monomial ideal (the intersection of s squares of monomial ideals).

We have the following theorem:

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Theorem 2.1: Let V,n,k be as in the previous section. Then: i) if k = 2 then [ν2(G2,n+1)]s

is defective for s < E(2, n + 1) = ⌊n+12 ⌋ with defectiveness 2s(s − 1); ii) while if k ≥ 3

and ks ≤ n + 1, then [νk(Gk,n+1)]s has the expected dimension.

Proof: The case k = 2 is known (e.g. see [E] or also [Z]) but we give the proof here for

the sake of completeness and also because the “monomial ideal” approach makes it quite

easy. First, assume that 2s ≤ n + 1. We have to consider the vector space

W = [(e0, e1)2 ∩ (e2, e3)2 ∩ ... ∩ (e2s−2, e2s−1)2]n−1

Once we note that (ei, ej)2 = (ei ∧ ej) we have that

W = (e0 ∧ e1 ∧ · · · ∧ e2s−2 ∧ e2s−1)n−1

Thus, W will trivially be = {0} if and only if n − 1 < 2s. This immediately gives that

E(2, n + 1) = ⌊n + 1

2⌋.

When n− 1 ≥ 2s, we get that a basis for W is given by the decomposable exterior vectors

of the form:

e0 ∧ e1 ∧ ... ∧ e2s−1 ∧ eα1∧ ... ∧ eαt

,

where t = n − 1 − 2s and α1, . . . , αt can be any t elements in {2s, . . . , n}. So,

dimK W = dimK(∧t〈e2s, ..., en〉) =

(n − 2s + 1

n − 2s − 1

)

=

(n − 2s + 1

2

)

.

A simple computation shows that this is not the expected dimension for W (the expected

dimension is(n+1

2

)− s(2n − 1) in this case) and that the defectiveness is δ = 2s2 − 2s =

2s(s − 1). This completes the proof of i).

As for ii), let k ≥ 3 and ks ≤ n + 1. Recall that we may always suppose that

k ≤ (n + 1)/2 (see Remark 2 before Definition 1.1).

Case 1: Suppose that 2s > n + 1 − k.

In order for there to be a monomial ei1 ∧ . . .∧ ein+1−kin W we must have at least two

of {i1, . . . , in+1−k} from each of the s subsets

{0, . . . , k − 1}, {k, . . . , 2k − 1}, · · · , {ks − k, . . . , ks − 1}.

-9- 2/1/08

So, if 2s > n + 1 − k this will automatically give that W = 0. It remains to check that,

under the given hypothesis on s, n and k, the expected dimension of W is also 0.

Observe that in order to have: k ≥ 3, ks ≤ n + 1 and 2s > n + 1 − k we must have:

n + 1 − k

2< s ≤ n + 1

k. (††)

This implies that

k2 − k(n + 1) + 2(n + 1) > 0 (‡)

The discriminant of the quadratic expression in (‡) is ∆ = (n + 1)(n − 7), so for

n < 7, the quadratic expression has the same sign for every k. Since the coefficient of k2

is positive this means that (‡) is always satisfied for such n.

Now, the conditions: k ≥ 3, n ≤ 6, ks ≤ 7, and 2k ≤ 7 has only one solution:

k = 3, s = 2, n = 5. A quick check shows that for these parameters the expected

dimension for W is indeed 0. This gives E(3, 6) = 2 and ν3(G3,6)2 = P14 as required.

When n = 7, the only possibility for k is 3 and there is no s satisfying (††).So, suppose that n ≥ 8. The quadratic equation associated to (‡) then has two distinct

roots

r1 =n + 1

2−

√∆

2<

n + 1

2+

√∆

2= r2

Thus (‡) can be satisfied only for k ≤ r1 and k ≥ r2. But, r2 > n+12

, so we need only

consider k ≤ r1.

Notice that, for n > 8, n+12

−√

∆2

< 3, so only case left to consider is n = 8 and k = 3

and there is no s satisfying (††) for those values of n and k.

This completes Case 1.

Case 2: 2s ≤ n + 1 − k.

In this case W is certainly 6= 0. Since W is the degree n + 1 − k part of a monomial

ideal, to compute dimK W it is enough to count all the monomials of degree n + 1 − k

which are NOT in W .

These are the monomials eα1∧ . . . ∧ eαn+1−k

with the property that, when we choose

α1, . . . , αn+1−k from {0, 1, . . . , n} we must make sure that from at least one of the s

subsets

{0, . . . , k − 1}, {k, . . . , 2k − 1}, · · · , {(s − 1)k, . . . , sk − 1}

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we have chosen either nothing or one index.

If we concentrate at first (say) on the subset {0, . . . , k − 1}, then we need to choose

n + 1 − k elements from {0, . . . , n} such that all n + 1 − k are outside {0, . . . , k − 1} or at

most one is in {0, , . . . , k − 1}Since there are exactly n+1−k elements outside {0, . . . , k−1}, we have only 1 choice

if we choose nothing from {0, . . . , k−1}. However, there are n+1−k subsets of {k, . . . , n}consisting of (n + 1 − k) − 1 elements and to those we can add any one of the k elements

in {0, . . . , k − 1}. This gives us a total of 1 + k(n + 1 − k) choices. Since we can do this

for each of the s subsets of k elements above, we get a total of s[k(n + 1− k) + 1] choices.

I.e. there are exactly s[k(n + 1 − k) + 1] monomial of degree n + 1 − k outside W . Since

this is precisely the expected codimension of W , the theorem is proved.

The previous result has the following immediate corollary:

Corollary 2.2: Let V, n, k be as above but suppose that k ≥ 3. Then all the chordal

varieties, νk(Gk,n+1)2 have the expected dimension.

3. Some Final Remarks

Since there appears to be so little in the literature about secant varieties to Grass-

mannians (but lots of folklore) we would like to collect some scattered results we have

found and record them in this section. This will include an (apparently) new example of

a deficient variety.

First, let m, n, d be three positive integers such that m = n + d − 1. Let V , W be

vector spaces over K such that dimK V = n and dimK W = m. Let φd : Pn−1 −→ P

t be

the Veronese embedding (and so t =(md

)− 1). The Plucker embedding of ∧dW is also in

Pt since, as Ehrenborg pointed out in [E], there is a 1-1 correspondence beetween d-subsets

of an m-set and d-multisubsets of an n-set.

This suggests that in looking for deficient Grassmannians we consider all the deficient

Veronese varieties. We do that now.

Case 1: All the quadratic Veronese varieties are defective, i.e. when d = 2 we can

choose n arbitrarily. What about the corresponding Grassmannians? For m = 4, 5 the

Grassmannian of lines is not defective, for trivial reasons. However, it is the case that all

the other Grassmannians of lines are defective. (See Theorem 2.1 i) ). This is hopeful.

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Case 2: There are exactly four other defective Veronese varieties. They correspond to:

i) d = 4, n = 3 and hence m = 6;

ii) d = 4, n = 4 and hence m = 7;

iii) d = 3, n = 5 and hence m = 7;

iv) d = 4, n = 5 and hence m = 8.

Now, i) suggests we look at ν4(G4,6). Since ∧4K6 ≃ ∧2K6, this is indeed defective,

it is (again) a Grassmannian of lines.

¿From ii) we consider ν4(G4,7). Since ∧4K7 ≃ ∧3K7 we should check if ν3(G3,7) is

defective. The secant plane variety should fill P34 but it does not, as we will now show.

By our method, we have to determine dimZ, where

Z = [(e0, e1, e2)2 ∩ (e3, e4, e5)2 ∩ (e6, v, w)2]4

where {e0, ..., e6} is a basis of W and v, w are generic vectors in W .

We can actually suppose that v,w ∈ 〈e0, ..., e5〉, in fact if v = e6 + v1 and w = e6 +w1

we get:

(e6, v, w)2 = (e6 ∧ v, e6 ∧ w, v ∧ w) = (e6 ∧ v1, e6 ∧ w1, v1 ∧ w1) = (e6, v1, w1)2.

Notice that:

Z ′ = [(e0, e1, e2)2 ∩ (e3, e4, e5)2]4 = 〈ei1 ∧ ei2 ∧ ei3 ∧ ei4〉4,

with i1 < i2 ∈ {0, 1, 2}, i3 < i4 ∈ {3, 4, 5}.

Thus, if there is something in Z it must be of the form v ∧ w ∧ Γ, Γ ∈ ∧2〈e0, ..., e5〉.Actually, by the genericity of v, w we can suppose that 〈e0, ..., e5〉 = 〈v, w, e1, ..., e4〉, hence

we can consider Γ ∈ 〈e1, ..., e4〉. We can even forther suppose that v = e0 + ... + e5 and

w =∑5

i=0 biei and then it is enough to insure that for each of the 6 monomials, m, in Γ,

the summands of v∧w∧m which are not in Z ′ are all = 0. This gives us 6 linear equations

in the b’s whose coefficient matrix is:

b3 − b0 b0 − b2 0 b1 − b0 0 0b4 − b0 0 b0 − b2 0 b1 − b0 0b5 − b0 0 0 0 0 0

0 0 0 0 0 b5 − b0

0 b5 − b4 b3 − b5 0 0 b5 − b1

0 0 0 b5 − b4 b3 − b5 b5 − b2

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This is a matrix whose rank is = 5 and hence we get dimZ = 1, i.e. ν3(G3,7) is defective

for secant P2’s, with defectivity 1.

Note: Apparently this example is well-known, as we recently learned

from J. Landsberg (private communication). We were unable to find

a reference to it in the literature.

From iii) we are again brought to consideer ν3(G3,7), which we have just done.

From,iv) we should consider ν4(G4,8). We now show that this is indeed a defective

variety.

The Grassmannian ν4(G4,8) lies in P69. Since dim ν4(G4,8) = 16 we have, by our

Corollary 2.2, that dim(ν4(G4,8))2 = 33, the expected dimension. One expects that

dim(ν4(G4,8))3 = 50, but we will show that (ν4(G4,8))3 = 49 instead.

By what we have seen before, it will be enough to prove that dimK W = 20, where

if H1, H2 and H3 are three generic subspaces of K8, each of dimension 4, with bases

{vi1, vi2, vi3, vi4}, i = 1, 2, 3, then

W = [(v11, v12, v13, v14)2 ∩ (v21, v22, v23, v24)2 ∩ (v31, v32, v33, v34)2]4

in the exterior algebra ∧(K8).

We can assume that H1 = 〈e0, e1, e2, e3〉, where the ei are part of a standard basis for

K8. Consider H3 ∩ 〈H2, ei〉. This is a one dimensional subspace of H3 which we’ll denote

by 〈wi〉 By the genericity of the subspaces we may suppose that {w0, w1, w2, w3} are a

basis for H3. But, wi = ei + ui for ui ∈ H2 and again, using the genericity, we can assume

that {u0, u1, u2, u3} are a basis for H2.

Putting all this together we can assume, without loss of generality, that

H1 = 〈e0, e1, e2, e3〉, H2 = 〈e4, e5, e6, e7〉 with {e0, . . . , e7} a basis for K8

and

H3 = 〈e0 + e4, e1 + e5, e2 + e6, e3 + e7〉.

Now a simple calculation using Macaulay 2 shows that dimK W is indeed 20 and we

are done. In fact, a simple hand check shows that the 20 forms in this space are:

e0 ∧ e1 ∧ e4 ∧ e5; e0 ∧ e2 ∧ e4 ∧ e6; e0 ∧ e3 ∧ e4 ∧ e7;

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e1 ∧ e2 ∧ e5 ∧ e6; e1 ∧ e3 ∧ e5 ∧ e7; e2 ∧ e3 ∧ e6 ∧ e7,

the 12 forms

ei ∧ ei+4 ∧ (ej ∧ ek+4 ± ek ∧ ej+4); i 6= j 6= k, i, j, k ∈ {0, 1, 2, 3}

and the two forms

(e0 ∧ e5 ± e1 ∧ e4) ∧ (e2 ∧ e7 ± e3 ∧ e6); (e0 ∧ e6 ± e2 ∧ e4) ∧ (e1 ∧ e6 ± e3 ∧ e5).

This shows that(ν4(G4,8))3 is 1-defective. One wonders about the dimensions of

the other secant varieties of ν4(G4,8). A calculation (using Macaulay 2) shows that

(ν4(G4,8))5 = P69. It is not hard to show that if we fix three general points P , Q, and R on

the Grassmannian in P69 then the linear system of hyperplanes which contain the tangent

spaces TP and TQ and which also contain R have exactly a fixed line in the tangent plane

TR. (This is the geometric reason why the secant planes to ν4(G4,8) are 1-deficient.)

Hence, when we take a fourth point S on ν4(G4,8) and consider the linear system of

hyperplanes on P69 which contains TP , TQ, and TR and which also contain S, that system

contains three fixed lines in TS . We expect those lines to be linearly independent and

hence that the 4-secants to ν4(G4,8) are 4-defective. Calculations with Macaulay 2 seem

to confirm that expectation.

It would be tempting, at this point, to conjecture that inasmuch as we have exhausted

the list of defective Veronese varieties then we have also exhausted the list of defective

Grassmannians! Indeed, we hoped that this might be so, but J. Landsberg informed us

that he had a communication from M. Catalano-Johnson asserting that ν3(G3,9) is also

defective.

Since we couldn’t find that example in the literature we provide a proof now. Notice

that, in view of Theorem 2.1 ii, the space of secant P2’s does have the correct dimension.

So, we will now show that (ν3(G3,9))4 has dimension 73 instead of the expected dimension

75.

The argument follows the same lines we used to find the defectivity of ν4(G4,8)3.

Following the discussion in § 2 we need to find dimK W , where if Hi, i = 1, . . . , 4 are 4

generics 3-dimensional subspaces of K9 and a basis for Hi is {vi1, vi2, vi3}, then

W =[

4⋂

i=1

(vi1, vi2, vi3)2]

6.

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It is easy to see that, with no loss of generality, we can assume the four subspaces are:

H1 = 〈e1, e2, e3〉, H2 = 〈e4, e5, e6〉, H3 = 〈e7, e8, e9〉

and

H4 = 〈e1 + e4 + e7, e2 + e5 + e8, e3 + e6 + e9〉.

Using the exterior algebra routines in Macaulay 2 we find that dimK W = 10 and so the

dimension of ν4(G(4, 8))3 is 73, as stated.

These last two examples suggested that we should check ν3(G3,12) and ν4(G4,12) as

well. We have verified that ν3(G3,12)5 and ν4(G4,12)4 are not defective.

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[W]: W.Wakeford. On canonical forms. Proc. London Math. Soc. 18 (1919/20). 403-410.

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M.V.Catalisano, Dip. Matematica, Univ. di Genova, Italy.

e-mail: catalisa@dima.unige.it

A.V.Geramita, Dept. Math. and Stats. Queens’ Univ. Kingston, Ont., Canada and

Dip. di Matematica, Univ. di Genova. Italy.

e-mail: geramita@dima.unige.it ; tony@mast.queensu.ca

A.Gimigliano, Dip. di Matematica and CIRAM, Univ. di Bologna, Italy.

e-mail: gimiglia@dm.unibo.it

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